Expression of type Forall

from the theory of proveit.core_expr_types.tuples

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import a, b, f, i, j, k, l
from proveit.core_expr_types.tuples import neg_shift_equiv_both
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Add, Integer, Natural, one, subtract

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [f], instance_expr = Forall(instance_param_or_params = [a, b], instance_expr = Forall(instance_param_or_params = [i, j, k, l], instance_expr = neg_shift_equiv_both, conditions = [InSet(subtract(Add(j, one), i), Natural), Equals(Add(i, a), Add(k, b)), Equals(Add(j, a), Add(l, b))]), domain = Integer))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{f}~\left[\forall_{a, b \in \mathbb{Z}}~\left[\forall_{i, j, k, l~|~\left(\left(j + 1\right) - i\right) \in \mathbb{N}, \left(i + a\right) = \left(k + b\right), \left(j + a\right) = \left(l + b\right)}~\left(\left(f\left(i + a\right), f\left(\left(i - 1\right) + a\right), \ldots, f\left(j + a\right)\right) = \left(f\left(k + b\right), f\left(\left(k - 1\right) + b\right), \ldots, f\left(l + b\right)\right)\right)\right]\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 11 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 61 body: 4
3	ExprTuple	61
4	Operation	operator: 11 operand: 6
5	ExprTuple	6
6	Lambda	parameters: 7 body: 8
7	ExprTuple	77, 79
8	Conditional	value: 9 condition: 10
9	Operation	operator: 11 operand: 14
10	Operation	operator: 25 operands: 13
11	Literal
12	ExprTuple	14
13	ExprTuple	15, 16
14	Lambda	parameters: 17 body: 18
15	Operation	operator: 34 operands: 19
16	Operation	operator: 34 operands: 20
17	ExprTuple	76, 74, 65, 66
18	Conditional	value: 21 condition: 22
19	ExprTuple	77, 23
20	ExprTuple	79, 23
21	Operation	operator: 37 operands: 24
22	Operation	operator: 25 operands: 26
23	Literal
24	ExprTuple	27, 28
25	Literal
26	ExprTuple	29, 30, 31
27	ExprTuple	32
28	ExprTuple	33
29	Operation	operator: 34 operands: 35
30	Operation	operator: 37 operands: 36
31	Operation	operator: 37 operands: 38
32	ExprRange	lambda_map: 39 start_index: 64 end_index: 40
33	ExprRange	lambda_map: 41 start_index: 42 end_index: 43
34	Literal
35	ExprTuple	44, 45
36	ExprTuple	46, 47
37	Literal
38	ExprTuple	48, 49
39	Lambda	parameter: 82 body: 50
40	Operation	operator: 80 operand: 74
41	Lambda	parameter: 82 body: 52
42	Operation	operator: 80 operand: 65
43	Operation	operator: 80 operand: 66
44	Operation	operator: 72 operands: 55
45	Literal
46	Operation	operator: 72 operands: 56
47	Operation	operator: 72 operands: 57
48	Operation	operator: 72 operands: 58
49	Operation	operator: 72 operands: 59
50	Operation	operator: 61 operand: 67
51	ExprTuple	74
52	Operation	operator: 61 operand: 68
53	ExprTuple	65
54	ExprTuple	66
55	ExprTuple	63, 64
56	ExprTuple	76, 77
57	ExprTuple	65, 79
58	ExprTuple	74, 77
59	ExprTuple	66, 79
60	ExprTuple	67
61	Variable
62	ExprTuple	68
63	Operation	operator: 72 operands: 69
64	Operation	operator: 80 operand: 76
65	Variable
66	Variable
67	Operation	operator: 72 operands: 71
68	Operation	operator: 72 operands: 73
69	ExprTuple	74, 75
70	ExprTuple	76
71	ExprTuple	78, 77
72	Literal
73	ExprTuple	78, 79
74	Variable
75	Literal
76	Variable
77	Variable
78	Operation	operator: 80 operand: 82
79	Variable
80	Literal
81	ExprTuple	82
82	Variable